For polyhedra, the notions shape and shape domination in the above definition can be replaced by the notions homotopy type and homotopy domination, respectively. Consequently, the capacity and depth in shape category of compacta are the same as in the homotopy category of CW-complexes and homotopy classes of cellular maps between them (see [1]}). S. Mather in [2]} proved that every polyhedron dominates only countably many different homotopy types (see also [3]}).
